ransp ort Prop erties - adv-res.com · T ransp ort Prop erties CO 2-enhanced coal-b ed methane reco v ery in olv es motion of a uid under the in uence of gradien ts of pressure, concen

Transport Properties

CO2-enhanced coal-bed methane recovery involvesmotion of a uid under the

in uence of gradients of pressure, concentration and temperature. In addition

to the equation of state, the viscosity, di�usivities and thermal conductivityof the uid can be expected to play a major role in the characterization of

this process. Calculation of these transport properties can be approached

either theoretically (mainly using the non-equilibrium kinetic theory formal-

ism developed by Chapman and Enskog), or semi-empirically (by application

of various corresponding-states principles). Both approaches have proved to

be useful. Thus, for low-density uids, the kinetic theory approach has the

advantage of greater rigor and generality, but the disadvantage of requiring

knowledge of the intermolecular potentials, which might not be available.

On the other hand, the corresponding-states-based correlations have provedparticularly valuable in the extrapolation to high densities.

Pure Gas Viscosity at Low Pressure

The viscosity of gases at low pressure can be determined from nonequilibriumkinetic theory according to a procedure devised by Chapman and Enskog.Molecular interaction e�ects are manifested in the values of the so-called`collision integrals', which involve the pairwise interaction potential, and ap-

pear as temperature-dependent correction factors to the elementary kinetic-

theory expressions for the transport properties. For all realistic models ofintermolecular potentials, these integrals require numerical evaluation, and

for the important particular case of the Lennard-Jones potential function,viz.,

(r) = 4�

"��

r

�12

��

r

�6#;

the values of the collision integral can be calculated by an empirical approxi-mant developed by Neufeld et al. [P.D. Neufeld, A.R. Janzen, and R.A. Aziz,

Journal of Chemical Physics, 57: 1100 (1972)], as a function of dimensionlesstemperature T � = kT=�. This is

v = A=T �B + C exp(�DT �) + E exp(�FT �);

where A = 1:16145, B = 0:14874, C = 0:52487, D = 0:77320, E = 2:16178,and F = 2:43787. The expression for the viscosity (in micropoise, 10�7 Pa�s)

1

is

� =5

16

p�MRT

��2v

= 26:69

pMT

��2v

;

where M is the molar mass (g�mol�1), and the length-scale parameter � is

expressed in �Angstr�om units (10�10 m). Since expressions for the viscosityof multi-component gas mixtures involve the viscosities of the pure gases,

this formula is best implemented by supplying an array of the molar masses

and Lennard-Jones parameters �, �=k [which are tabulated in Appendix C of

the monograph \The Properties of Gases and Liquids", by R.C. Reid, J.M.

Prausnitz, and T.K. Sherwood, 3ed., New York: McGraw-Hill (1977)] for the

components, as follows:

SUBROUTINE ETACEN(NC,EPSK,ETA,RM,SIGMA,TK)

C This subroutine returns an array of low-pressure gas viscosities

C calculated from the Chapman-Enskog formula, in which the collision

C integral is estimated from the Lennard-Jones parameters

C $\epsilon/k$ in EPSK(1:NC) and $\sigma$ in SIGMA(1:NC) according

C to the empirical correlation devised by P.D. Neufeld, A.R. Janzen,

C and R.A. Aziz, J. Chem. Phys., 57: 1100 (1972).

C

C Reference: R.C. Reid, J.M. Prausnitz and T.K. Sherwood, "The

C Properties of Gases and Liquids", 3rd edition, New York: McGraw-Hill

C (1977), pp. 395-399.

C

IMPLICIT DOUBLE PRECISION(A-H,O-Z)

DIMENSION EPSK(NC) !Lennard-Jones parameters $\epsilon/k$, K

DIMENSION ETA(NC) !Viscosity, micropoise

DIMENSION RM(NC) !Molar masses, g/mol

DIMENSION SIGMA(NC) !Lennard-Jones diameters $\sigma$, Angstrom

C

C Parameters in the correlation for OMEGAV as a function of

C dimensionless temperature TSTAR:

C

C \[

C \Omega_v = A/T^{*B}+C\exp(-D T^{*})+E\exp(-F T^{*})

C \]

C

PARAMETER(RGAS=8.31451)

PARAMETER(A=1.16145)

PARAMETER(B=0.14874)

PARAMETER(C=0.52487)

PARAMETER(D=0.77320)

PARAMETER(E=2.16178)

2

PARAMETER(F=2.43787)

CONST=26.69

DO 1 I=1,NC

TSTAR=TK/EPSK(I)

SIGMSQ=SIGMA(I)*SIGMA(I)

OMEGAV=A/TSTAR**B+C*DEXP(-D*TSTAR)+E*DEXP(-F*TSTAR)

ETA(I)=CONST*DSQRT(RM(I)*TK)/OMEGAV/SIGMA(I)/SIGMA(I)

1 CONTINUE

RETURN

END

Mixed Gas Viscosity at Low Pressure

The kinetic theory of gases can also be used to obtain an expression for the

low-pressure viscosity of a gas mixture. According to the discussion given byReid et al. (1977), this is of the form

�m =NXi=1

yi�iPN

j=1�ijyj

;

where the array of mixture coeÆcients �ij can be determined according toa variety of approximations. The simplest of these, due to Herning and

Zipperer (1936) [F. Herning and L. Zipperer,Gas Wasserfach, 79: 49 (1936)],is

�ij =

sMj

Mi

;

while according to Wilke (1950) [C.R. Wilke, Journal of Chemical Physics,18: 517 (1950)],

�ij =[1 + (�i=�j)

1=2(Mj=Mi)1=4]2

[8(1 +Mi=Mj)]1=2:

These expressions work best for mixtures of nonpolar gases, characterized by

relatively weak intereractions, and as such seem to be suitable for applicationto the study of carbon dioxide-methane mixtures. Brokaw [R.S. Brokaw,Industrial and Engineering Chemistry Process Design and Development, 8:

240 (1969)] devised a more elaborate model which is applicable to mixtures

containing polar components:

�ij = AijSij

s�i

�j

3

where

Aij =mijqMij

"1 +

Mij �M0:45

ij

2(1 +Mij) + (1 +M0:45

ij)m�0:5

ij=(1 +mij)

#;

Mij =Mi

Mj

; mij =

"4

(1 +Mij)(1 + 1=Mij)

#0:25

;

and the polar correction factor Sij is set equal to 1 if the species are non-

polar. The viscosity of the low-pressure mixture is evaluated according to

these models by the function ETALPM:

FUNCTION ETALPM(MODEL,NC,ETA,RM,Y)

C This subroutine determines the low-pressure viscosity of a gas

C mixture containing NC components with individual viscosities

C ETA(1:NC), molar masses RM(1:NC), and mole fractions Y(1:NC)

C


PARAMETER(MAXC=10) !Maximum number of components

C

C Adjustable arrays

C

DIMENSION ETA(NC) !Pure-component viscosities, micropoise


DIMENSION Y(NC) !Mole fractions

C

C Fixed arrays

C

DIMENSION PHI(MAXC*MAXC) !Mixing coefficients, phi

DIMENSION PHIY(MAXC) !Elements of PHI*Y

C

C Identification of invalid inputs

C

IF(NC.GT.MAXC) THEN

WRITE(6,*) 'Error return from ETALPM: Too many components'

WRITE(6,*) NC,MAXC

STOP

ENDIF

C

C Construction of the matrix of parameters PHI

C

DO 2 I=1,NC

DO 1 J=1,NC

IJ=NC*(I-1)+J

4

ETAIJ=ETA(I)/ETA(J)

RMIJ=RM(I)/RM(J)

IF(MODEL.EQ.1) THEN

C

C In model 1, the mixing coefficients are obtained as described by

C C.R. Wilke, J. Chem. Phys., 18: 517 (1950):

C

XUPPER=1.+DSQRT(ETAIJ/DSQRT(RMIJ))

XUPPER=XUPPER*XUPPER

XLOWER=DSQRT(8.*(1.+RMIJ))

PHI(IJ)=XUPPER/XLOWER

ELSEIF(MODEL.EQ.2) THEN

C


C F. Herning and L. Zipperer, Gas Wasserfach, 79: 49 (1936):

C

PHI(IJ)=1./DSQRT(RMIJ)


C


C R.S. Brokaw, Ind. Eng. Chem. Proc. Des. Dev., 8: 240 (1969), for

C mixtures of non-polar gases:

C

SIJ=1. !Polar correction factor = 1

RMMIJ=(4./(1.+RMIJ)/(1.+1./RMIJ))**0.25

F1=RMIJ-RMIJ**0.45

F2=2.*(1.+RMIJ)

F3=(1.+RMIJ**0.45)/DSQRT(RMMIJ)/(1.+RMMIJ)

AIJ=(1.+F1/(F2+F3))*RMMIJ/DSQRT(RMIJ)

PHI(IJ)=DSQRT(ETAIJ)*SIJ*AIJ

ELSE

WRITE(6,*) 'Error return from ETALPM:'

WRITE(6,*) 'Unknown option ',MODEL

ENDIF

1 CONTINUE

2 CONTINUE

C

C Product of the arrays PHI and Y

C

DO 4 I=1,NC

PHIY(I)=0.

DO 3 J=1,NC

IJ=NC*(I-1)+J

PHIY(I)=PHIY(I)+PHI(IJ)*Y(J)

3 CONTINUE

5

4 CONTINUE

C

C Application of the mixing rule

C

ETALPM=0.

DO 5 I=1,NC

ETALPM=ETALPM+Y(I)*ETA(I)/PHIY(I)

5 CONTINUE

RETURN

END

The operation of both of these codes can be illustrated by Example 9.5 of

Reid et al. (1977, page 414), which involves calculating the viscosity of a mix-

ture of methane (1) and butane, with y1=0.697, at 20ÆC. FromAppendix C of

Reid et al. (1977), the Lennard-Jones parameters are �=k=148.6 K, �=3.758�A for methane, and �=k=531.4 K, �=4.687 �A for butane. The choice be-tween the three approximations for the mixture coeÆcients is expressed bythe value of the parameter MODEL. For the Wilke expression (MODEL=1),the pure-component viscosities are 109.68 �P for methane and 72.92 �P for

butane, which compare favorably with the experimental values 109.4 and72.74 obtained by Kestin and Yata [J. Kestin and J. Yata, Journal of Chem-

ical Physics, 49: 4780 (1968)]. The calculated mixture value is 92.48, whichis in reasonable agreement with the experimental value 93.35 obtained bythe same authors; the discrepancy is a little greater than 1%. An example

of a driver program is as follows:

C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12

C Calculation of viscosities of gaseous mixtures of methane (1)

C and butane (2)

C


DIMENSION EPSK(2),ETA(2),RM(2),SIGMA(2),Y(2)

NC=2

TK=293.15

C

C Lennard-Jones parameters and molar masses

C

EPSK(1)=148.6

SIGMA(1)=3.758

EPSK(2)=531.4

SIGMA(2)=4.687

RM(1)=16.043

RM(2)=58.124

6

C

C Pure-component viscosities

C

CALL ETACEN(NC,EPSK,ETA,RM,SIGMA,TK)

WRITE(6,'(T2,A24,F6.2,A11)')

,'Component 1 viscosity = ',ETA(1),' micropoise'

WRITE(6,'(T2,A24,F6.2,A11)')


C

C Comparison of models for mixture viscosity

C

Y(1)=0.697

Y(2)=0.303

DO 1 MODEL=1,3

TEST=ETALPM(MODEL,NC,ETA,RM,Y)

WRITE(6,'(T2,A6,I1,A21,F6.2,A11)')

, 'Model ',MODEL,' Mixture viscosity = ',TEST,' micropoise'

1 CONTINUE

END

This produces the output:

Component 1 viscosity = 109.68 micropoise


Model 1 Mixture viscosity = 92.48 micropoise



Gas Viscosities at High Pressure and Density

The method recommended by Reid et al. for calculating the e�ect of pressure

on the viscosity of a gas mixture is that of Dean and Stiel [D.E. Dean andL.I. Stiel, AIChE Journal, 11: 526 (1965)], which is based on the use of a

corresponding-states principle to determine a residual viscosity, or additive

correction to the low-pressure viscosity. This in turn requires an accurate

equation of state, together with a suitable method for estimating the pseudo-critical parameters for the mixture. The formula is

(�m � �Æm)�m = 1:08[exp(1:439�rm)� exp(�1:11�1:858

rm)];

where the superscript Æ is used to distinguish the low-pressure viscosity from

that at the pressure of interest. The reduced density of the mixture is ob-tained by dividing the actual density by the pseudocritical density (obtained

7

from the Prausnitz-Gunn mixture rules), and the parameter � is

�m =T 1=6

rm

M1=2

m p2=3

rm

where the mixture reduced pressure prm and temperature Trm are similarly

obtained from the pseudocritical parameters, but the mixture molar mass

Mm is a simple mole-fraction average. The Prausnitz-Gunn mixture rules

are implemented by the following subroutine PGRULE:

SUBROUTINE PGRULE(NC,TC,PC,RHOC,RM,TPC,RHOPC,PPC,RMAV,Y)

C This subroutine implements the Prausnitz-Gunn mixture rules for

C the pseudocritical temperature TPC, density RHOPC, and pressure

C PPC, of an NC-component fluid mixture characterized by mole

C fractions Y(1:NC), molar masses RM(1:NC), critical pressures

C PC(1:NC), and critical densities RHOC(1:NC).

C


PARAMETER(RGAS=8.31451) !Gas constant, J/K/mol

DIMENSION PC(NC) !Critical pressures, MPa

DIMENSION RHOC(NC) !Critical densities, g/cm^3


DIMENSION TC(NC) !Critical temperatures, K


RMAV=0.

TPC=0.

ZPC=0.

VPC=0.

DO 1 I=1,NC

RMAV=RMAV+Y(I)*RM(I)

TPC=TPC+Y(I)*TC(I)

VPC=VPC+Y(I)*RM(I)/RHOC(I)

ZPC=ZPC+Y(I)*PC(I)*RM(I)/(RGAS*TC(I)*RHOC(I))

1 CONTINUE

PPC=ZPC*RGAS*TPC/VPC

RETURN

END

From the account of the Dean-Stiel method given by Reid et al. (1977), it

is evident that the crucial input is not in fact the pressure, but the densityof the mixture - and, of course, its composition, which appears not only inthe low-pressure viscosity but also in the pseudocritical constants through

the Prausnitz-Gunn mixture rules. This suggests that a subprogram based

8

on this model should accept as arguments the temperature, the number of

components, and adjustable arrays containing the mole fractions, critical

parameters, and molar masses of the components. Within this subprogram,

the �rst step would be to calculate the low-pressure mixture viscosity, the

second step would be to calculate the mixture pseudocritical parameters, and

�nally the mixture viscosity at the pressure of interest would be calculated.

The existing function ETALPM that evaluates the low-pressure viscosity

of gas mixtures consists of the integer parameters MODEL, NC, and the NC-

dimensional arrays ETA, RM, and Y; the pure-gas low-pressure viscosities

ETA are determined separately by calling the subroutine ETACEN. The cal-culation of pressure/density e�ects is more easily performed by a subroutine

ETAMIX that accepts the additional inputs arrays of critical parameters PC,

RHOC, TC, for the components, as well as the temperature TK and mixture

density RHOGC3 (g�cm�3). This arrangement allows for maximum exibil-

ity, in that the required density can be derived from a call to any suitable

equation of state. It is also useful to include not only the low- (or zero-)pressure viscosity ETA0 in addition to the viscosity ETARHO correspondingto the density RHOGC3 returned by the equation of state, since this makesit easier to identify the pressures at which appreciable deviations from the

low-pressure values are expected.In the example given by Reid et al. (op. cit., p. 434), the required density

was in fact obtained from the corresponding-states correlation developed byPitzer, but use of a speci�c equation of state would almost certainly produceresults of superior accuracy. Since volumetric data for the ternary system

methane-carbon dioxide-nitrogen have not been reported in suÆcient numberto give reliable estimates of the mixture second and third virial coeÆcients,

this mode of application can be illustrated instead by use of the three-termvirial equation of state

z = 1 +B�+ C�2 z3 � z2 � �z � = 0;

for the ternary system methane (1)-ethane (2)-carbon dioxide (3), where

B =3Xi=1

3Xj=1

Bijxixj; C =3Xi=1

3Xj=1

3Xk=1

Cijkxixjxk:

Values of the mixture virial coeÆcients Bij , Cijk have been derived by Houet al. [H. Hou, J.C. Holste, K.R. Hall, K.N. Marsh, and B.E. Gammon,

9

Journal of Chemical and Engineering Data, 41: 344-353 (1996)] from Burnet-

isochoric measurements at 300 and 320 K. Instead of using the exact solution

of the equivalent cubic equation for z,

z3 � z2 � �z � = 0;

where � � Bp=RT , � C(p=RT )2, the density for speci�ed pressure can be

obtained more expeditiously by Newton-Raphson iteration for the equivalent

equation

f(z) = 1 +�

z+

z2� z; f 0(z) = �

�

z2�

2

z3� 1;

which avoids the need for a call to another subroutine to solve the equation.

The FORTRAN code for this subroutine, and a driver program that tests it

at the state point p=6.36283 MPa, T=320 K, x1=0.24756, x2=0.56013 (forwhich the experimental value of the compressibility factor is z=0.71454) isas follows:

C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12

C Equation of state for mixtures of methane (1), ethane (2), and

C carbon dioxide (3)

C


PARAMETER(RMW1=16.0428,RMW2=30.06964,RMW3=44.0098)

PMPA=6.36283

TK=320.

X1=0.24756

X2=0.56013

CALL VEOS23(PMPA,RHOGC3,TK,X1,X2)

END

SUBROUTINE VEOS23(PMPA,RHOGC3,TK,X1,X2)

C Given temperature TK (K), pressure PMPA (MPa), and mole fractions

C X1, X2 of methane and ethane, respectively, this subroutine

C returns the mass density RHOGC3 (mol cm^-3) of a ternary mixture

C of methane (1), ethane (2), and carbon dioxide (3)

C


PARAMETER(RGAS=8.31451)

PARAMETER(ITMAX=10,TOL=1.D-15)

DIMENSION B(3,3),C(3,3,3),RM(3),X(3)

C

C Molar masses, g mol^-1

C

10

DATA RM(1)/16.0428/

DATA RM(2)/30.0696/

DATA RM(3)/44.0098/

C

C Mixture 2nd virial coefficients (cm^3 mol^-1) reported by H. Hou,

C J.C. Holste, K.R. Hall, K.N. Marsh, and B.E. Gammon, J. Chem.

C Eng. Data, 41: 344-353 (1996)

C

DATA B(1,1)/ -35.17/

DATA B(1,2)/ -76.71/

DATA B(1,3)/ -54.02/

DATA B(2,1)/ -76.71/

DATA B(2,2)/-159.42/

DATA B(2,3)/-105.65/

DATA B(3,1)/ -54.02/

DATA B(3,2)/-105.65/

DATA B(3,3)/-104.54/

C

C Mixture 3rd virial coefficients reported by Hou et al. (1996),

C cm^6 mol^-2

C

DATA C(1,1,1)/2229./

DATA C(2,2,2)/9692./

DATA C(3,3,3)/4411./

DATA C(1,1,2)/3594./

DATA C(1,2,1)/3594./

DATA C(2,1,1)/3594./

DATA C(1,1,3)/2641./

DATA C(1,3,1)/2641./

DATA C(3,1,1)/2641./

DATA C(1,2,2)/5902./

DATA C(2,1,2)/5902./

DATA C(2,2,1)/5902./

DATA C(1,3,3)/3195./

DATA C(3,1,3)/3195./

DATA C(3,3,1)/3195./

DATA C(2,2,3)/6963./

DATA C(2,3,2)/6963./

DATA C(3,2,2)/6963./

DATA C(2,3,3)/5075./

DATA C(3,2,3)/5075./

DATA C(3,3,2)/5075./

DATA C(1,2,3)/4204./

DATA C(2,3,1)/4204./

DATA C(3,1,2)/4204./

11

DATA C(1,3,2)/4204./

DATA C(2,1,3)/4204./

DATA C(3,2,1)/4204./

C

C Second and third virial coefficients of the mixed fluid;

C average molar mass

C

X(1)=X1

X(2)=X2

X(3)=1.-X1-X2

BMIX=0.

CMIX=0.

RMAV=0.

DO 3 I=1,3

RMAV=RMAV+X(I)*RM(I)

DO 2 J=1,3

BMIX=BMIX+B(I,J)*X(I)*X(J)

DO 1 K=1,3

CMIX=CMIX+C(I,J,K)*X(I)*X(J)*X(K)

1 CONTINUE

2 CONTINUE

3 CONTINUE

C

C Newton-Raphson iteration for the compressibility factor

C

PRT=PMPA/RGAS/TK

BETA=PRT*BMIX

GAMMA=PRT*PRT*CMIX

Z=1.

DO 4 ITER=1,ITMAX

F=(GAMMA/Z+BETA)/Z+1.-Z

FPRIME=-1.-BETA/Z/Z-2.*GAMMA/Z/Z/Z

DZ=-F/FPRIME

RELERR=DABS(DZ/Z)

C WRITE(6,*) ITER,Z,RELERR

IF(RELERR.LT.TOL) GOTO 5

Z=Z+DZ

4 CONTINUE

WRITE(6,*) 'Warning: no convergence in ',ITMAX,' iterations'

5 RHOGC3=RMAV*PRT/Z

RETURN

END

It produces the output

1 1.00000000000000 0.266590971881958

12

2 0.733409028118042 2.574907815503465D-02

3 0.714524421733423 2.085336958233519D-04

4 0.714375419315002 1.338264719687369D-08

5 0.714375409754768 1.152368903920541D-16

which indicates that the iteration from z=1 gives satisfactory convergence.

There is good agreement between this value of z and the experimental value,

which indicates that the values assigned to the virial coeÆcients are correct.

Listings of subroutine ETAMIX and its driver program are as follows,

C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12

C Program to investigate pressure effects on the viscosity of

C multicomponent gas mixtures

C



DIMENSION PC(3),RHOC(3),TC(3)

C

C Lennard-Jones parameters and molar masses for methane (1),

C ethane (2), and carbon dioxide (3)

C

DATA EPSK(1),SIGMA(1),RM(1),Y(1)/148.6,3.758,16.0428,0.24756/



C

C Critical constants for mixture components

C

DATA PC(1),RHOC(1),TC(1)/4.54,0.1621,190.4/

DATA PC(2),RHOC(2),TC(2)/4.82,0.2032,305.2/

DATA PC(3),RHOC(3),TC(3)/7.28,0.4682,304.2/

NC=3

TK=320.

PMPA=6.36283

X1=Y(1)

X2=Y(2)

C

C Pure-component low-pressure viscosities

C


WRITE(6,'(T2,A24,F6.2,A11)')


WRITE(6,'(T2,A24,F6.2,A11)')


WRITE(6,'(T2,A24,F6.2,A11)')


13

C

C Mass density of gas mixture

C

CALL VEOS23(PMPA,RHOGC3,TK,X1,X2)

WRITE(6,'(T2,A10,E10.4,A8)') 'Density = ',RHOGC3,' g cm^-3'

C

C Comparison of models for low-pressure mixture viscosity

C

WRITE(6,'(T2,A5,2A20)') 'Model','eta0/micropoise','eta/micropoise'

DO 1 MODEL=1,3

CALL ETAMIX(MODEL,NC,ETA,ETA0,ETARHO,PC,RHOC,RHOGC3,RM,TC,Y)

WRITE(6,'(T2,I5,2F20.1)') MODEL,ETA0,ETARHO

1 CONTINUE

END

SUBROUTINE ETAMIX(MODEL,NC,ETA,ETA0,ETARHO,PC,RHOC,RHOGC3,RM,TC,Y)

C This subroutine determines the viscosity of a gas of mass density

C RHOGC3 (g cm^-3) containing NC components with individual

C viscosities ETA(1:NC), molar masses RM(1:NC), and mole fractions

C Y(1:NC)

C


PARAMETER(MAXC=10) !Maximum number of components

C

C Adjustable arrays

C

DIMENSION ETA(NC) !Pure-component viscosities, micropoise

DIMENSION PC(NC) !Critical pressures, MPa

DIMENSION RHOC(NC) !Critical densities, g cm^-3

DIMENSION RM(NC) !Molar masses, g mol^-1

DIMENSION TC(NC) !Critical temperatures, K


C

C Fixed arrays

C

DIMENSION PHI(MAXC*MAXC) !Mixing coefficients, phi

DIMENSION PHIY(MAXC) !Elements of PHI*Y

C

C Identification of invalid inputs

C

IF(NC.GT.MAXC) THEN

WRITE(6,*) 'Error return from ETAMIX: Too many components'

WRITE(6,*) NC,MAXC

STOP

ENDIF

C

14

C Construction of the matrix of parameters PHI

C

DO 2 I=1,NC

DO 1 J=1,NC

IJ=NC*(I-1)+J

ETAIJ=ETA(I)/ETA(J)

RMIJ=RM(I)/RM(J)

IF(MODEL.EQ.1) THEN

C


C C.R. Wilke, J. Chem. Phys., 18: 517 (1950):

C

XUPPER=1.+DSQRT(ETAIJ/DSQRT(RMIJ))

XUPPER=XUPPER*XUPPER

XLOWER=DSQRT(8.*(1.+RMIJ))

PHI(IJ)=XUPPER/XLOWER


C


C F. Herning and L. Zipperer, Gas Wasserfach, 79: 49 (1936):

C

PHI(IJ)=1./DSQRT(RMIJ)


C


C R.S. Brokaw, Ind. Eng. Chem. Proc. Des. Dev., 8: 240 (1969), for

C mixtures of non-polar gases:

C

SIJ=1. !Polar correction factor = 1

RMMIJ=(4./(1.+RMIJ)/(1.+1./RMIJ))**0.25

F1=RMIJ-RMIJ**0.45

F2=2.*(1.+RMIJ)

F3=(1.+RMIJ**0.45)/DSQRT(RMMIJ)/(1.+RMMIJ)

AIJ=(1.+F1/(F2+F3))*RMMIJ/DSQRT(RMIJ)

PHI(IJ)=DSQRT(ETAIJ)*SIJ*AIJ

ELSE

WRITE(6,*) 'Error return from ETAMIX:'

WRITE(6,*) 'Unknown option ',MODEL

ENDIF

1 CONTINUE

2 CONTINUE

C

C Product of the arrays PHI and Y

C

DO 4 I=1,NC

15

PHIY(I)=0.

DO 3 J=1,NC

IJ=NC*(I-1)+J

PHIY(I)=PHIY(I)+PHI(IJ)*Y(J)

3 CONTINUE

4 CONTINUE

C

C Application of the mixing rule

C

ETA0=0.

DO 5 I=1,NC

ETA0=ETA0+Y(I)*ETA(I)/PHIY(I)

5 CONTINUE

C

C Calculation of the pseudocritical constants and pseudoreduced

C density

C

CALL PGRULE(NC,TC,PC,RHOC,RM,TPC,RHOPC,PPC,RMAV,Y)

XI=TPC**(1./6.)/DSQRT(RMAV)/(PPC/0.101325)**(2./3.)

RHOR=RHOGC3/RHOPC

C

C Calculation of the pressure effects according to the correlation

C of D.E. Dean and L.I. Stiel, AIChE Journal, 11: 526 (1965)

C

ETARHO=ETA0+(DEXP(1.439*RHOR)-DEXP(-1.111*(RHOR**1.858)))*1.08/XI

RETURN

END

The results




Density = 0.9801E-01 g cm^-3

Model eta0/micropoise eta/micropoise

1 116.9 149.1

2 118.2 150.4

3 116.5 148.8

show that the additive correction to the low-pressure viscosity is of the rightorder, in comparison to the example given by Reid et al. and, more impor-tantly, that the e�ect of pressure (or density) on viscosity is not negligible.

This pressure e�ect introduces a further nonlinearity into the equation of

motion of the uid through the porous medium.

16

Comparison with Experiment: Viscosity

Several general observations can be made concerning the agreement be-

tween experimental measurements and the various models and correlations

described in the preceding sections.

The most fundamental of these is that the spherically-symmetric Lennard-

Jones potential, upon which the Neufeld correlation for the Chapman-Enskogcollision integrals is based, is not necessarily appropriate for all uids. In

uids containing molecules of irregular shape, the multipole moments associ-

ated with the unsymmetrical distribution of electronic charge density make

signi�cant contributions to the intermolecular potential. Inclusion of these

e�ects in the calculation of transport coeÆcients is in general complicated

by the tensorial character of multipolar interactions, and by the need for

explicit inclusion of the rotational coordinates in the integrations. Interac-

tions between molecules containing permanent dipole moments can describedby the Stockmayer potential function; as a practical matter, the use of theStockmayer potential or some more general function involving higher-ordermultipole tensors can be expected to a�ect not only the values of the col-lision integrals, but also their temperature-dependence. Thus, a degree of

skepticism is appropriate when using the Lennard-Jones potential, by way ofthe Neufeld correlation or some similar result, as the basis for predicting thetemperature dependence of the viscosity of molecular uids. In this regardit is signi�cant that Reid et al. give no details of the viscosity data fromwhich their compilation of Lennard-Jones parameters were obtained, or of

the data-reduction procedure used for this purpose, particularly in regard tothe weighting scheme used.

A second point is that the predictions of the low-pressure viscosity for

mixtures depend not only on theoretical justi�cation of the various mix-

ture models, but are limited ultimately by the method used to estimatethe required end-member viscosities. Thus, a more accurate result might

be obtained if the pure-component viscosities were experimentally measuredrather than obtained from a correlation. This e�ect could be signi�cant if the

temperature of interest were outside the temperature range represented in

the viscosity data from which the Lennard-Jones parameters were obtained.The accuracy of the pure- uid viscosity values obtained from ETACEN are

ultimately determined not only by the range of conditions in the originaldata, but also by the precision, and the regression procedure used to infer

the Lennard-Jones parameters.

17

By extension, the calculation of viscosities at high pressures is likewise

limited by the mixture-model used to determine the low-pressure viscosity, in

addition to the theoretical justi�cation of the corresponding-states relation-

ship (including the method used to estimate the pseudocritical parameters)

used to extrapolate to the conditions of interest, and on the accuracy of the

equation of state.

In the light of the foregoing observations, it is of interest to compare the

predictions of the three mixture models with low-pressure viscosity measure-

ments by Jackson [W.M. Jackson, \Viscosities of the binary gas mixtures,

methane-carbon dioxide and ethylene-argon", Journal of Physical Chemistry,60: 789 (1956)] for CH4-CO2 mixtures:

C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12

C Calculation of viscosities of gaseous mixtures of methane (1)

C and carbon dioxide (2)

C



DIMENSION RESID(3),SSR(3),TEST(3)

C


C

DATA EPSK(1),SIGMA(1),RM(1)/148.6,3.758,16.0428/


C

C Experimental data:

C [W.M. Jackson, Journal of Physical Chemistry, 60:789 (1956)]

C

DIMENSION XCH4(14),ETAEXP(14)

DATA XCH4( 1),ETAEXP( 1)/0.000,151.0/

DATA XCH4( 2),ETAEXP( 2)/0.022,150.9/

DATA XCH4( 3),ETAEXP( 3)/0.103,149.4/

DATA XCH4( 4),ETAEXP( 4)/0.183,147.7/

DATA XCH4( 5),ETAEXP( 5)/0.297,145.3/

DATA XCH4( 6),ETAEXP( 6)/0.421,141.4/

DATA XCH4( 7),ETAEXP( 7)/0.537,137.3/

DATA XCH4( 8),ETAEXP( 8)/0.651,132.8/

DATA XCH4( 9),ETAEXP( 9)/0.730,129.1/

DATA XCH4(10),ETAEXP(10)/0.789,124.8/

DATA XCH4(11),ETAEXP(11)/0.850,122.6/

DATA XCH4(12),ETAEXP(12)/0.905,119.2/

DATA XCH4(13),ETAEXP(13)/0.933,117.4/

DATA XCH4(14),ETAEXP(14)/1.000,111.4/

NC=2

18

TK=298.15

NDATA=14

C

C Pure-component viscosities

C


WRITE(6,'(T2,A24,F6.2,A11)')


WRITE(6,'(T2,A24,F6.2,A11)')


C

C Comparison of models for mixture viscosity

C

SSR(1)=0.

SSR(2)=0.

SSR(3)=0.

WRITE(6,'(T2,A5,8A8)') 'XCH4','eta/uP','Model 1','Res.',

,'Model 2','Res.','Model 3','Res.'

DO 2 I=1,NDATA

Y(1)=XCH4(I)

Y(2)=1.-XCH4(I)

DO 1 MODEL=1,3

TEST(MODEL)=ETALPM(MODEL,NC,ETA,RM,Y)

RESID(MODEL)=TEST(MODEL)-ETAEXP(I)

SSR(MODEL)=SSR(MODEL)+RESID(MODEL)*RESID(MODEL)

1 CONTINUE

WRITE(6,'(T2,F5.3,8F8.1)') XCH4(I),ETAEXP(I),

, (TEST(K),RESID(K),K=1,3)

2 CONTINUE

DO 3 MODEL=1,3

SDR=DSQRT(SSR(MODEL)/DBLE(NDATA-1))

WRITE(6,'(T2,A6,I1,A31,F4.2)')

, 'Model ',MODEL,' Residual standard deviation = ',SDR

3 CONTINUE

END

From the output



XCH4 eta/uP Model 1 Res. Model 2 Res. Model 3 Res.

0.000 151.0 151.0 0.0 151.0 0.0 151.0 0.0

0.022 150.9 150.6 -0.3 150.4 -0.5 150.5 -0.4

0.103 149.4 149.1 -0.3 148.4 -1.0 148.5 -0.9

0.183 147.7 147.4 -0.3 146.2 -1.5 146.3 -1.4

19

0.297 145.3 144.6 -0.7 142.9 -2.4 143.0 -2.3

0.421 141.4 141.0 -0.4 138.9 -2.5 138.9 -2.5

0.537 137.3 136.9 -0.4 134.6 -2.7 134.6 -2.7

0.651 132.8 132.2 -0.6 129.9 -2.9 129.9 -2.9

0.730 129.1 128.4 -0.7 126.4 -2.7 126.3 -2.8

0.789 124.8 125.3 0.5 123.5 -1.3 123.3 -1.5

0.850 122.6 121.7 -0.9 120.2 -2.4 120.1 -2.5

0.905 119.2 118.2 -1.0 117.2 -2.0 117.0 -2.2

0.933 117.4 116.2 -1.2 115.5 -1.9 115.4 -2.0

1.000 111.4 111.3 -0.1 111.3 -0.1 111.3 -0.1

Model 1 Residual standard deviation = 0.64



the measured pure-component viscosities are seen to be in excellent agree-

ment with those produced by ETACEN, apart from the discrepancy of 0.1�P for the viscosity of methane. This is less than 0.1%, and well within theprecision of 0.2% claimed by the author. (Incidentally, the Orsat gas analysisprocedure was identi�ed as the dominant contribution to the experimentalerror.) The deviations from experiment - as expressed quantitatively by the

standard deviations - can therefore be attributed almost entirely to the mix-ture models. The best overall predictions are seen to be those from the Wilkecorrelation (Model 1), which, somewhat surprisingly, are signi�cantly betterthan the predictions of the more elaborate model of Brokaw (Model 3). It isalso noteworthy that all three models underestimate the mixture viscosity.

The performance of the Dean-Stiel correlation can be compared withmeasured viscosities from DeWitt and Thodos [K. J. De Witt and G. Thodos,\Viscosities of Binary Mixtures in the Dense Gaseous State: The Methane-Carbon Dioxide System," Canadian Journal of Chemical Engineering, 44:148-151 (1966)] at pressures up to about 680 atm. The Hou et al. three-

term virial equation of state cannot be used to provide the required densityhere, since the upper pressure limit is far higher than the pressure range

over which that equation was parameterized, but fortunately, DeWitt and

Thodos recorded the uid density, as well as the pressure, corresponding totheir viscosity measurements. The viscosities of a mixture containing 75.7mol% carbon dioxide up to 683 atm and 200ÆC, are in the following program,

in which the low-pressure viscosity is estimated from the Wilke correlation

(MODEL=1):

C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12

C Program to investigate pressure effects on the viscosity of

20

C multicomponent gas mixtures

C



DIMENSION PC(2),RHOC(2),TC(2)

C

C Low-Temperature viscosity of methane-carbon dioxide mixtures

C K.J. DeWitt and G. Thodos, Canadian Journal of Chemical

C Engineering, 44: 148 (1966)

C

DIMENSION TCELS(4),ETAEX0(4),PATM(4,11),RHO(4,11),ETAEXP(4,11)

DATA TCELS(1),ETAEX0(1)/50.1,155.1/

DATA PATM(1, 1),RHO(1, 1),ETAEXP(1, 1)/ 33.41,0.0543,162.6/


DATA PATM(1, 3),RHO(1, 3),ETAEXP(1, 3)/135.55,0.3370,279.1/







DATA PATM(1,10),RHO(1,10),ETAEXP(1,10)/613.64,0.7763,821.6/


DATA TCELS(2),ETAEX0(2)/100.4,176.4/












DATA TCELS(3),ETAEX0(3)/150.7,197.4/










21



DATA TCELS(4),ETAEX0(4)/200.4,216.6/












C

C Lennard-Jones parameters and molar masses for methane (1),

C ethane (2), and carbon dioxide (3)

C



C

C Critical constants for mixture components

C

DATA PC(1),RHOC(1),TC(1)/4.54,0.1621,190.4/

DATA PC(2),RHOC(2),TC(2)/7.28,0.4682,304.2/

NC=2

XCO2=0.757

Y(2)=XCO2

Y(1)=1.-XCO2

MODEL=1

WRITE(6,'(T2,A7,F5.3)') 'xCO2 = ',XCO2

DO 2 ITEMP=1,4

TK=273.15+TCELS(ITEMP)

C

C Pure-component low-pressure viscosities

C


WRITE(6,'(T2,A2,F5.1,A10)') 'T=',TCELS(ITEMP),' degrees C'

WRITE(6,'(T2,A24,F6.2,A11)')

, 'Component 1 viscosity = ',ETA(1),' micropoise'

WRITE(6,'(T2,A24,F6.2,A11)')

, 'Component 2 viscosity = ',ETA(2),' micropoise'

WRITE(6,'(T2,A24,F6.2,A11)')

, 'Low-pressure viscosity = ',ETAEX0(ITEMP),' micropoise'

WRITE(6,'(T2,A6,4A14)') 'p/atm','d/g cm^-3','eta(calc.)/uP',

22

, 'eta(expt.)/uP','Residual/uP'

DO 1 I=1,11

PMPA=PATM(ITEMP,I)*0.101325

RHOGC3=RHO(ITEMP,I)

CALL ETAMIX(MODEL,NC,ETA,ETA0,ETARHO,PC,RHOC,

, RHOGC3,RM,TC,Y)

RESID=ETARHO-ETAEXP(ITEMP,I)

WRITE(6,'(T2,F6.2,F14.4,3F14.2)') PATM(ITEMP,I),

, RHO(ITEMP,I),ETARHO,ETAEXP(ITEMP,I),RESID

1 CONTINUE

WRITE(6,*)

2 CONTINUE

END

The output

xCO2 = 0.757

T= 50.1 degrees C

Component 1 viscosity =119.06 micropoise


Low-pressure viscosity =155.10 micropoise

p/atm d/g cm^-3 eta(calc.)/uP eta(expt.)/uP Residual/uP

33.41 0.0543 167.37 162.60 4.77

68.66 0.1254 186.75 180.10 6.65

135.55 0.3370 283.61 279.10 4.51

203.39 0.5126 426.43 419.80 6.63

273.07 0.6055 542.27 532.30 9.97

340.77 0.6609 631.44 613.60 17.84

410.18 0.7000 705.86 675.80 30.06

477.89 0.7298 770.06 726.10 43.96

546.61 0.7562 833.02 781.10 51.92

613.64 0.7763 885.18 821.60 63.58

682.02 0.7908 925.26 855.60 69.66

T=100.4 degrees C





34.64 0.0447 186.23 182.40 3.83

67.50 0.0930 198.11 192.20 5.91

137.04 0.2145 241.02 236.70 4.32

205.63 0.3416 307.45 303.00 4.45

273.41 0.4436 381.86 375.50 6.36

341.11 0.5173 452.44 444.70 7.74

23

409.84 0.5731 518.60 509.50 9.10

478.57 0.6152 577.63 565.90 11.73

545.59 0.6483 630.57 614.90 15.67

613.30 0.6788 685.20 662.50 22.70

683.38 0.7022 731.35 701.30 30.05

T=150.7 degrees C





34.16 0.0409 205.20 202.70 2.50

68.79 0.0789 214.12 209.70 4.42

138.00 0.1686 242.39 237.70 4.69

205.29 0.2586 281.12 277.70 3.42

273.07 0.3424 327.73 322.50 5.23

340.77 0.4127 376.63 370.40 6.23

408.48 0.4715 426.40 419.00 7.40

478.23 0.5188 473.84 465.90 7.94

545.59 0.5582 519.47 511.20 8.27

614.66 0.5926 564.65 556.00 8.65

681.68 0.6219 607.63 596.40 11.23

T=200.4 degrees C





33.41 0.0328 222.08 221.00 1.08

69.81 0.0700 230.41 227.00 3.41

138.75 0.1425 251.67 247.40 4.27

205.84 0.2140 279.13 276.10 3.03

273.41 0.2828 311.99 308.10 3.89

341.79 0.3465 348.86 343.80 5.06

409.50 0.3993 385.04 379.20 5.84

478.91 0.4432 419.85 413.70 6.15

545.25 0.4830 455.82 449.40 6.42

614.32 0.5199 493.58 485.90 7.68

682.70 0.5508 528.97 521.00 7.97

shows that the correlation consistently overestimates the viscosity by an ex-

tent that increases with increasing density (as one would expect), but de-

creases with increasing temperature. Over the range of conditions (up to

100 atm) relevant to coal-bed methane technology, the deviations are about3% at the lowest temperature (50ÆC) in the measurements of DeWitt and

24

Thodos; following this trend, the deviations can be expected to be somewhat

larger at lower temperatures.

The tentative conclusions that can be drawn from the calculations pre-

sented here are that (1) the Wilke correlation gives the best results for the

low-pressure mixture viscosity, and (2) the Dean-Stiel method gives results

of satisfactory accuracy for pressures up to about 100 atm.

Di�usion CoeÆcients

The starting point of the discussion of di�usion coeÆcients presented by Reid

et al. is the Chapman-Enskog formula

DAB =3

16

sM1 +M2

M1M2

2�kT1

n��2AB

D

fD =3

16

s2�kT

�

1

n��2AB

D

fD

where m1, m1 are the molecular masses, D is the collision integral appro-priate for di�usivity, and fD is a dimensionless parameter with a value close

to 1. The subscript applied to D signi�es di�usion of A in B, or vice versa.If the molar density is assumed to be obtained from the ideal gas equationof state in the form p = NkT=V = nkT , this becomes

DAB =3

16

sM1 +M2

M1M2

2�kT1

n��2AB

D

fD =3

16

s2�kT

�

kT

p��2AB

D

fD

Expressing the reduced mass � in molar units requires inclusion of a factorof Avogadro's number NA in the numerator of the fraction under the square

root, and if pressure is expressed in atmospheres and the length-scale of theinteraction potential function in �Angstr�om units, the conversion factor is

3

16

s2000�kNA

�

1020k

101325�= 1:858670 � 10�7m2 � s�1

�(g �mol�1)1=2(atm)(�A)2(K)�3=2

The working equation for the di�usivity is therefore

DAB=m2 � s�1 = [1:858670 � 10�7(g �mol�1)1=2(atm)(�A)2(K)�3=2]

�T 3=2fD

p��2AB

D

sM1 +M2

M1M2

25

whereM1,M1 are the molar masses (g�mol�1). This is in agreement with the

formula 11.3-2 given by Reid et al., apart from the trivial factor of 1� 104

required to convert m2s�1 to cm2s�1.

The di�usion collision integral can be also be calculated, as a function of

the Lennard-Jones potential parameters, from an empirical approximation

developed by Neufeld et al., viz.,

d = A=T �B + C exp(�DT �) + E exp(�FT �) +G exp(�HT �):

Here, the reduced temperature T � for the pair AB is calculated from a char-

acteristic temperature obtained from a geometric mean mixing rule, and a

length scale obtained from an arithmetic mean:

�AB

k=

p�A�B

k�AB =

�A + �B

2:

Subroutine DCCEN0, which returns a matrix of di�usivities calculated inthis manner, is as follows:

SUBROUTINE DCCEN0(NC,DCOEFF,EPSK,PBAR,RM,SIGMA,TK)

C This subroutine returns an array of low-pressure gas diffusivities

C calculated from the Chapman-Enskog formula, in which the collision

C integral is estimated from the Lennard-Jones parameters

C $\epsilon/k$ in EPSK(1:NC) and $\sigma$ in SIGMA(1:NC) according

C to the empirical correlation devised by P.D. Neufeld, A.R. Janzen,

C and R.A. Aziz, J. Chem. Phys., 57: 1100 (1972).

C

C Reference: R.C. Reid, J.M. Prausnitz and T.K. Sherwood, "The

C Properties of Gases and Liquids", New York: McGraw-Hill (1977),

C 3ed., pp. 549-550.

C


DIMENSION EPSK(NC) !Lennard-Jones parameters $\epsilon/k$, K

DIMENSION DCOEFF(NC*NC)!Diffusivity, cm^2 s^-1


DIMENSION SIGMA(NC) !Lennard-Jones diameters $\sigma$, Angstrom

C

C Parameters in the correlation for OMEGAV as a function of

C dimensionless temperature TSTAR:

C

C \[

C \Omega_d = A/T^{*B}+C\exp(-D T^{*})+E\exp(-F T^{*})

C +G\exp(-H T^{*})

C \]

26

C

PARAMETER(PIE=3.14159265358979324)

PARAMETER(A=1.06036)

PARAMETER(B=0.15610)

PARAMETER(C=0.19300)

PARAMETER(D=0.47635)

PARAMETER(E=1.03587)

PARAMETER(F=1.52996)

PARAMETER(G=1.76474)

PARAMETER(H=3.89411)

PARAMETER(CONST=1.858670D-03)

PARAMETER(FD=1.D+00)

PATM=PBAR/1.01325

DO 2 I=1,NC

DO 1 J=1,NC

IJ=NC*(I-1)+J

EPSKIJ=DSQRT(EPSK(I)*EPSK(J))

SIGMIJ=(SIGMA(I)+SIGMA(J))/2.

RMUIJ=RM(I)*RM(J)/(RM(I)+RM(J))

TSTAR=TK/EPSKIJ

OMEGAD=A/TSTAR**B+C*DEXP(-D*TSTAR)+E*DEXP(-F*TSTAR)

, +G*DEXP(-H*TSTAR)

DCOEFF(IJ)=FD*CONST*(TK**1.5)/DSQRT(RMUIJ)

, /(PATM*SIGMIJ*SIGMIJ*OMEGAD)

1 CONTINUE

2 CONTINUE

RETURN

END

The driver program

C23456789 123456789 123456789 123456789 123456789 123456789 123456789 12

C Calculation of diffusivities of gaseous mixtures

C


DIMENSION EPSK(2),RM(2),SIGMA(2),DCOEFF(4)

NC=2

TK=590.

C


C

SIGMA(1)=3.941

SIGMA(2)=3.798

EPSK(1)=195.2

EPSK(2)=71.4

27

RM(1)=28.013

RM(2)=44.010

PBAR=1.01325

CALL DCCEN0(NC,DCOEFF,EPSK,PBAR,RM,SIGMA,TK)

DO 2 I=1,NC

DO 1 J=1,NC

IJ=NC*(I-1)+J

WRITE(6,*) DCOEFF(IJ)

1 CONTINUE

2 CONTINUE

END

calculates the binary di�usion coeÆcient of nitrogen and carbon dioxide,

and produces an array containing the self-di�usion coeÆcients of the pure

endmembers as well:

1 1 0.483

1 2 0.510

2 1 0.510

2 2 0.514

The value of the o�-diagonal element is in good agreement with that obtainedin Example 11-1 from Reid et al.

28

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ransp ort Prop erties - adv-res.com · T ransp ort Prop erties CO 2-enhanced coal-b ed methane reco v ery in olv es motion of a uid under the in uence of gradien ts of pressure, concen